Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 74.0% → 99.2%

Time: 9.7s

Alternatives: 11

Speedup: 0.4×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (* J_s (if (<= t_1 (- INFINITY)) (- U_m) (if (<= t_1 4e+290) t_1 U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_1 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J_m) * t_0) * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U_m
	elif t_1 <= 4e+290:
		tmp = t_1
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= 4e+290)
		tmp = t_1;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U_m;
	elseif (t_1 <= 4e+290)
		tmp = t_1;
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, 4e+290], t$95$1, U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 43.3

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.00000000000000025e290

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   if 4.00000000000000025e290 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 15.2%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 15.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 58.3

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 58.4%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.4%

       \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ t_2 := \cos \left(0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+141}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0)))))
        (t_2 (* (cos (* 0.5 K)) (* J_m -2.0))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -2e+141)
        t_2
        (if (<= t_1 -4e-114)
          (* (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0)) (* -2.0 J_m))
          (if (<= t_1 5e+217) t_2 U_m)))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double t_2 = cos((0.5 * K)) * (J_m * -2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -2e+141) {
		tmp = t_2;
	} else if (t_1 <= -4e-114) {
		tmp = sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else if (t_1 <= 5e+217) {
		tmp = t_2;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	t_2 = Float64(cos(Float64(0.5 * K)) * Float64(J_m * -2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -2e+141)
		tmp = t_2;
	elseif (t_1 <= -4e-114)
		tmp = Float64(sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	elseif (t_1 <= 5e+217)
		tmp = t_2;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -2e+141], t$95$2, If[LessEqual[t$95$1, -4e-114], N[(N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+217], t$95$2, U$95$m]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 43.3

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.00000000000000003e141 or -4.0000000000000002e-114 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.00000000000000041e217

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{1}{2} \cdot K\right), \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right), \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \color{blue}{\left(\frac{-1}{2} \cdot K\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \color{blue}{\frac{\frac{-1}{4} \cdot {U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \color{blue}{\frac{\frac{\frac{-1}{4} \cdot {U}^{2}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \color{blue}{\frac{\frac{\frac{-1}{4} \cdot {U}^{2}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\color{blue}{\frac{\frac{-1}{4} \cdot {U}^{2}}{J}}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\color{blue}{\frac{-1}{4} \cdot {U}^{2}}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \color{blue}{\left(U \cdot U\right)}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \color{blue}{\left(U \cdot U\right)}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      16. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)}\right) \]

      20. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(-0.5 \cdot K\right), \frac{\frac{-0.25 \cdot \left(U \cdot U\right)}{J}}{\cos \color{blue}{\left(-0.5 \cdot K\right)}}\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot J, \cos \left(-0.5 \cdot K\right), \frac{\frac{-0.25 \cdot \left(U \cdot U\right)}{J}}{\cos \left(-0.5 \cdot K\right)}\right)} \]

   6. Taylor expanded in J around inf

      \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.5%

      \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot -2\right)} \]

   if -2.00000000000000003e141 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -4.0000000000000002e-114

   1. Initial program 99.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot \left(-2 \cdot J\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot \left(-2 \cdot J\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \cdot \left(-2 \cdot J\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + 1}} \cdot \left(-2 \cdot J\right) \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4} \cdot {U}^{2}}{{J}^{2}}} + 1} \cdot \left(-2 \cdot J\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{4} \cdot {U}^{2}}{\color{blue}{J \cdot J}} + 1} \cdot \left(-2 \cdot J\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4}}{J} \cdot \frac{{U}^{2}}{J}} + 1} \cdot \left(-2 \cdot J\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{{U}^{2}}{J}, 1\right)}} \cdot \left(-2 \cdot J\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{J}}, \frac{{U}^{2}}{J}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \color{blue}{\frac{{U}^{2}}{J}}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{\color{blue}{U \cdot U}}{J}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{\color{blue}{U \cdot U}}{J}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      14. lower-*.f64 62.9

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \color{blue}{\left(-2 \cdot J\right)} \]

   5. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)} \]

   if 5.00000000000000041e217 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 33.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 33.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 49.2

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.3%

       \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+165}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J\_m \cdot -2\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 -2e+165)
        (* (cos (* 0.5 K)) (* J_m -2.0))
        (if (<= t_2 5e+217)
          (* t_1 (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0)))
          U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= -2e+165) {
		tmp = cos((0.5 * K)) * (J_m * -2.0);
	} else if (t_2 <= 5e+217) {
		tmp = t_1 * sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= -2e+165)
		tmp = Float64(cos(Float64(0.5 * K)) * Float64(J_m * -2.0));
	elseif (t_2 <= 5e+217)
		tmp = Float64(t_1 * sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, -2e+165], N[(N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+217], N[(t$95$1 * N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 43.3

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1.9999999999999998e165

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in U around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{1}{2} \cdot K\right), \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot J}, \cos \left(\frac{1}{2} \cdot K\right), \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      4. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right), \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \color{blue}{\left(\frac{-1}{2} \cdot K\right)}, \frac{-1}{4} \cdot \frac{{U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \color{blue}{\frac{\frac{-1}{4} \cdot {U}^{2}}{J \cdot \cos \left(\frac{1}{2} \cdot K\right)}}\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \color{blue}{\frac{\frac{\frac{-1}{4} \cdot {U}^{2}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \color{blue}{\frac{\frac{\frac{-1}{4} \cdot {U}^{2}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\color{blue}{\frac{\frac{-1}{4} \cdot {U}^{2}}{J}}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\color{blue}{\frac{-1}{4} \cdot {U}^{2}}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \color{blue}{\left(U \cdot U\right)}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \color{blue}{\left(U \cdot U\right)}}{J}}{\cos \left(\frac{1}{2} \cdot K\right)}\right) \]

      16. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}\right) \]

      18. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(\frac{-1}{2} \cdot K\right), \frac{\frac{\frac{-1}{4} \cdot \left(U \cdot U\right)}{J}}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)}\right) \]

      20. lower-*.f64 64.1

          \[\leadsto \mathsf{fma}\left(-2 \cdot J, \cos \left(-0.5 \cdot K\right), \frac{\frac{-0.25 \cdot \left(U \cdot U\right)}{J}}{\cos \color{blue}{\left(-0.5 \cdot K\right)}}\right) \]

   5. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot J, \cos \left(-0.5 \cdot K\right), \frac{\frac{-0.25 \cdot \left(U \cdot U\right)}{J}}{\cos \left(-0.5 \cdot K\right)}\right)} \]

   6. Taylor expanded in J around inf

      \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.5%

      \[\leadsto \cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot -2\right)} \]

   if -1.9999999999999998e165 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.00000000000000041e217

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]

      2. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + 1}} \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{4} \cdot {U}^{2}}{{J}^{2}}} + 1} \]

      4. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\frac{\frac{1}{4} \cdot {U}^{2}}{\color{blue}{J \cdot J}} + 1} \]

      5. times-frac N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{\frac{1}{4}}{J} \cdot \frac{{U}^{2}}{J}} + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{{U}^{2}}{J}, 1\right)}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{J}}, \frac{{U}^{2}}{J}, 1\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \color{blue}{\frac{{U}^{2}}{J}}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{\color{blue}{U \cdot U}}{J}, 1\right)} \]

      10. lower-*.f64 79.7

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{\color{blue}{U \cdot U}}{J}, 1\right)} \]

   5. Applied rewrites 79.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)}} \]

   if 5.00000000000000041e217 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 33.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 33.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 49.2

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.3%

       \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(\frac{J\_m}{U\_m} \cdot J\_m\right) \cdot -2 - U\_m\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(J\_m \cdot \left(0.25 \cdot K\right), K, J\_m \cdot -2\right) \cdot 1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (- (* (* (/ J_m U_m) J_m) -2.0) U_m))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+269)
      t_0
      (if (<= t_2 -4e+102)
        (* (fma (* J_m (* 0.25 K)) K (* J_m -2.0)) 1.0)
        (if (<= t_2 -1e-224) t_0 U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (((J_m / U_m) * J_m) * -2.0) - U_m;
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -2e+269) {
		tmp = t_0;
	} else if (t_2 <= -4e+102) {
		tmp = fma((J_m * (0.25 * K)), K, (J_m * -2.0)) * 1.0;
	} else if (t_2 <= -1e-224) {
		tmp = t_0;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(Float64(Float64(J_m / U_m) * J_m) * -2.0) - U_m)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+269)
		tmp = t_0;
	elseif (t_2 <= -4e+102)
		tmp = Float64(fma(Float64(J_m * Float64(0.25 * K)), K, Float64(J_m * -2.0)) * 1.0);
	elseif (t_2 <= -1e-224)
		tmp = t_0;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - U$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+269], t$95$0, If[LessEqual[t$95$2, -4e+102], N[(N[(N[(J$95$m * N[(0.25 * K), $MachinePrecision]), $MachinePrecision] * K + N[(J$95$m * -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, -1e-224], t$95$0, U$95$m]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e269 or -3.99999999999999991e102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-224

   1. Initial program 71.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} + -1 \cdot U} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot -2} + -1 \cdot U \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({J}^{2} \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)} \cdot -2 + -1 \cdot U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{J}^{2} \cdot \left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot -2\right)} + -1 \cdot U \]

      4. *-commutative N/A

         \[\leadsto {J}^{2} \cdot \color{blue}{\left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)} + -1 \cdot U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({J}^{2} \cdot -2\right) \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}} + -1 \cdot U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({J}^{2} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{J}^{2} \cdot -2}, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot J\right)} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot J\right)} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \color{blue}{\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}}, -1 \cdot U\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{\color{blue}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}{U}, -1 \cdot U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}^{2}}{U}, -1 \cdot U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}^{2}}{U}, -1 \cdot U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}}^{2}}{U}, -1 \cdot U\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \color{blue}{\left(\frac{-1}{2} \cdot K\right)}}^{2}}{U}, -1 \cdot U\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(\frac{-1}{2} \cdot K\right)}^{2}}{U}, \color{blue}{\mathsf{neg}\left(U\right)}\right) \]

      18. lower-neg.f64 30.3

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(-0.5 \cdot K\right)}^{2}}{U}, \color{blue}{-U}\right) \]

   5. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(-0.5 \cdot K\right)}^{2}}{U}, -U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \frac{{J}^{2}}{U} - \color{blue}{U} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \frac{J \cdot J}{U} \cdot -2 - \color{blue}{U} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.2%

       \[\leadsto \left(\frac{J}{U} \cdot J\right) \cdot -2 - U \]

   if -2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999991e102

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 86.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot J + \color{blue}{\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}}\right) \cdot 1 \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2} + -2 \cdot J\right)} \cdot 1 \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)} + -2 \cdot J\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left({K}^{2} \cdot J\right)} + -2 \cdot J\right) \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot {K}^{2}\right) \cdot J} + -2 \cdot J\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot {K}^{2}\right) \cdot J + -2 \cdot J\right) \cdot 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot {K}^{2}, J, -2 \cdot J\right)} \cdot 1 \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4}} \cdot {K}^{2}, J, -2 \cdot J\right) \cdot 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot {K}^{2}}, J, -2 \cdot J\right) \cdot 1 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

      13. lower-*.f64 50.3

          \[\leadsto \mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

   8. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, J \cdot -2\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto \mathsf{fma}\left(J \cdot \left(0.25 \cdot K\right), \color{blue}{K}, J \cdot -2\right) \cdot 1 \]

   if -1e-224 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 66.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 66.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 35.3

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.3%

       \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \left(\frac{J\_m}{U\_m} \cdot J\_m\right) \cdot -2 - U\_m\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(\left(-2 \cdot J\_m\right) \cdot t\_1\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_1}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\left(J\_m \cdot \mathsf{fma}\left(0.25 \cdot K, K, -2\right)\right) \cdot 1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (- (* (* (/ J_m U_m) J_m) -2.0) U_m))
        (t_1 (cos (/ K 2.0)))
        (t_2
         (*
          (* (* -2.0 J_m) t_1)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_1)) 2.0))))))
   (*
    J_s
    (if (<= t_2 -2e+269)
      t_0
      (if (<= t_2 -4e+102)
        (* (* J_m (fma (* 0.25 K) K -2.0)) 1.0)
        (if (<= t_2 -1e-224) t_0 U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = (((J_m / U_m) * J_m) * -2.0) - U_m;
	double t_1 = cos((K / 2.0));
	double t_2 = ((-2.0 * J_m) * t_1) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_1)), 2.0)));
	double tmp;
	if (t_2 <= -2e+269) {
		tmp = t_0;
	} else if (t_2 <= -4e+102) {
		tmp = (J_m * fma((0.25 * K), K, -2.0)) * 1.0;
	} else if (t_2 <= -1e-224) {
		tmp = t_0;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = Float64(Float64(Float64(Float64(J_m / U_m) * J_m) * -2.0) - U_m)
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(Float64(-2.0 * J_m) * t_1) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_1)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -2e+269)
		tmp = t_0;
	elseif (t_2 <= -4e+102)
		tmp = Float64(Float64(J_m * fma(Float64(0.25 * K), K, -2.0)) * 1.0);
	elseif (t_2 <= -1e-224)
		tmp = t_0;
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(J$95$m / U$95$m), $MachinePrecision] * J$95$m), $MachinePrecision] * -2.0), $MachinePrecision] - U$95$m), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, -2e+269], t$95$0, If[LessEqual[t$95$2, -4e+102], N[(N[(J$95$m * N[(N[(0.25 * K), $MachinePrecision] * K + -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$2, -1e-224], t$95$0, U$95$m]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e269 or -3.99999999999999991e102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-224

   1. Initial program 71.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} + -1 \cdot U} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot -2} + -1 \cdot U \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({J}^{2} \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)} \cdot -2 + -1 \cdot U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{J}^{2} \cdot \left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot -2\right)} + -1 \cdot U \]

      4. *-commutative N/A

         \[\leadsto {J}^{2} \cdot \color{blue}{\left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)} + -1 \cdot U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({J}^{2} \cdot -2\right) \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}} + -1 \cdot U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({J}^{2} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{J}^{2} \cdot -2}, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot J\right)} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot J\right)} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \color{blue}{\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}}, -1 \cdot U\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{\color{blue}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}{U}, -1 \cdot U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}^{2}}{U}, -1 \cdot U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}^{2}}{U}, -1 \cdot U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}}^{2}}{U}, -1 \cdot U\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \color{blue}{\left(\frac{-1}{2} \cdot K\right)}}^{2}}{U}, -1 \cdot U\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(\frac{-1}{2} \cdot K\right)}^{2}}{U}, \color{blue}{\mathsf{neg}\left(U\right)}\right) \]

      18. lower-neg.f64 30.3

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(-0.5 \cdot K\right)}^{2}}{U}, \color{blue}{-U}\right) \]

   5. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(-0.5 \cdot K\right)}^{2}}{U}, -U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \frac{{J}^{2}}{U} - \color{blue}{U} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.3%

      \[\leadsto \frac{J \cdot J}{U} \cdot -2 - \color{blue}{U} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.2%

       \[\leadsto \left(\frac{J}{U} \cdot J\right) \cdot -2 - U \]

   if -2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999991e102

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 86.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot J + \color{blue}{\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}}\right) \cdot 1 \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2} + -2 \cdot J\right)} \cdot 1 \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)} + -2 \cdot J\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left({K}^{2} \cdot J\right)} + -2 \cdot J\right) \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot {K}^{2}\right) \cdot J} + -2 \cdot J\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot {K}^{2}\right) \cdot J + -2 \cdot J\right) \cdot 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot {K}^{2}, J, -2 \cdot J\right)} \cdot 1 \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4}} \cdot {K}^{2}, J, -2 \cdot J\right) \cdot 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot {K}^{2}}, J, -2 \cdot J\right) \cdot 1 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

      13. lower-*.f64 50.3

          \[\leadsto \mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

   8. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, J \cdot -2\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto \left(J \cdot \color{blue}{\mathsf{fma}\left(0.25 \cdot K, K, -2\right)}\right) \cdot 1 \]

   if -1e-224 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 66.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 66.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 35.3

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.3%

       \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\left(J\_m \cdot \mathsf{fma}\left(0.25 \cdot K, K, -2\right)\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-224}:\\ \;\;\;\;\left(J\_m \cdot J\_m\right) \cdot \frac{-2}{U\_m} - U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -2e+269)
      (- U_m)
      (if (<= t_1 -4e+102)
        (* (* J_m (fma (* 0.25 K) K -2.0)) 1.0)
        (if (<= t_1 -1e-224) (- (* (* J_m J_m) (/ -2.0 U_m)) U_m) U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -2e+269) {
		tmp = -U_m;
	} else if (t_1 <= -4e+102) {
		tmp = (J_m * fma((0.25 * K), K, -2.0)) * 1.0;
	} else if (t_1 <= -1e-224) {
		tmp = ((J_m * J_m) * (-2.0 / U_m)) - U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -2e+269)
		tmp = Float64(-U_m);
	elseif (t_1 <= -4e+102)
		tmp = Float64(Float64(J_m * fma(Float64(0.25 * K), K, -2.0)) * 1.0);
	elseif (t_1 <= -1e-224)
		tmp = Float64(Float64(Float64(J_m * J_m) * Float64(-2.0 / U_m)) - U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+269], (-U$95$m), If[LessEqual[t$95$1, -4e+102], N[(N[(J$95$m * N[(N[(0.25 * K), $MachinePrecision] * K + -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-224], N[(N[(N[(J$95$m * J$95$m), $MachinePrecision] * N[(-2.0 / U$95$m), $MachinePrecision]), $MachinePrecision] - U$95$m), $MachinePrecision], U$95$m]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e269

   1. Initial program 37.0%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 41.2

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{-U} \]

   if -2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999991e102

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 86.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot J + \color{blue}{\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}}\right) \cdot 1 \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2} + -2 \cdot J\right)} \cdot 1 \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)} + -2 \cdot J\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left({K}^{2} \cdot J\right)} + -2 \cdot J\right) \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot {K}^{2}\right) \cdot J} + -2 \cdot J\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot {K}^{2}\right) \cdot J + -2 \cdot J\right) \cdot 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot {K}^{2}, J, -2 \cdot J\right)} \cdot 1 \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4}} \cdot {K}^{2}, J, -2 \cdot J\right) \cdot 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot {K}^{2}}, J, -2 \cdot J\right) \cdot 1 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

      13. lower-*.f64 50.3

          \[\leadsto \mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

   8. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, J \cdot -2\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto \left(J \cdot \color{blue}{\mathsf{fma}\left(0.25 \cdot K, K, -2\right)}\right) \cdot 1 \]

   if -3.99999999999999991e102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-224

   1. Initial program 99.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} + -1 \cdot U} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{J}^{2} \cdot {\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot -2} + -1 \cdot U \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\left({J}^{2} \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)} \cdot -2 + -1 \cdot U \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{{J}^{2} \cdot \left(\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U} \cdot -2\right)} + -1 \cdot U \]

      4. *-commutative N/A

         \[\leadsto {J}^{2} \cdot \color{blue}{\left(-2 \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}\right)} + -1 \cdot U \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left({J}^{2} \cdot -2\right) \cdot \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}} + -1 \cdot U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({J}^{2} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{J}^{2} \cdot -2}, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot J\right)} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(J \cdot J\right)} \cdot -2, \frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \color{blue}{\frac{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}{U}}, -1 \cdot U\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{\color{blue}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}{U}, -1 \cdot U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}^{2}}{U}, -1 \cdot U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}}^{2}}{U}, -1 \cdot U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}}^{2}}{U}, -1 \cdot U\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)}^{2}}{U}, -1 \cdot U\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \color{blue}{\left(\frac{-1}{2} \cdot K\right)}}^{2}}{U}, -1 \cdot U\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(\frac{-1}{2} \cdot K\right)}^{2}}{U}, \color{blue}{\mathsf{neg}\left(U\right)}\right) \]

      18. lower-neg.f64 27.9

          \[\leadsto \mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(-0.5 \cdot K\right)}^{2}}{U}, \color{blue}{-U}\right) \]

   5. Applied rewrites 27.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(J \cdot J\right) \cdot -2, \frac{{\cos \left(-0.5 \cdot K\right)}^{2}}{U}, -U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto -2 \cdot \frac{{J}^{2}}{U} - \color{blue}{U} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.9%

      \[\leadsto \frac{J \cdot J}{U} \cdot -2 - \color{blue}{U} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.9%

       \[\leadsto \left(J \cdot J\right) \cdot \frac{-2}{U} - U \]

   if -1e-224 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 66.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 66.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 35.3

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.3%

       \[\leadsto U \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{+102}:\\ \;\;\;\;\left(J\_m \cdot \mathsf{fma}\left(0.25 \cdot K, K, -2\right)\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-224}:\\ \;\;\;\;-U\_m\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 -2e+269)
      (- U_m)
      (if (<= t_1 -4e+102)
        (* (* J_m (fma (* 0.25 K) K -2.0)) 1.0)
        (if (<= t_1 -1e-224) (- U_m) U_m))))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -2e+269) {
		tmp = -U_m;
	} else if (t_1 <= -4e+102) {
		tmp = (J_m * fma((0.25 * K), K, -2.0)) * 1.0;
	} else if (t_1 <= -1e-224) {
		tmp = -U_m;
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -2e+269)
		tmp = Float64(-U_m);
	elseif (t_1 <= -4e+102)
		tmp = Float64(Float64(J_m * fma(Float64(0.25 * K), K, -2.0)) * 1.0);
	elseif (t_1 <= -1e-224)
		tmp = Float64(-U_m);
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, -2e+269], (-U$95$m), If[LessEqual[t$95$1, -4e+102], N[(N[(J$95$m * N[(N[(0.25 * K), $MachinePrecision] * K + -2.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, -1e-224], (-U$95$m), U$95$m]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -2.0000000000000001e269 or -3.99999999999999991e102 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-224

   1. Initial program 71.4%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 33.7

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 33.7%

      \[\leadsto \color{blue}{-U} \]

   if -2.0000000000000001e269 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -3.99999999999999991e102

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 86.7%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]

   6. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-2 \cdot J + \color{blue}{\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}}\right) \cdot 1 \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} \cdot J\right) \cdot {K}^{2} + -2 \cdot J\right)} \cdot 1 \]

      3. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)} + -2 \cdot J\right) \cdot 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \color{blue}{\left({K}^{2} \cdot J\right)} + -2 \cdot J\right) \cdot 1 \]

      5. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot {K}^{2}\right) \cdot J} + -2 \cdot J\right) \cdot 1 \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot {K}^{2}\right) \cdot J + -2 \cdot J\right) \cdot 1 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot {K}^{2}, J, -2 \cdot J\right)} \cdot 1 \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4}} \cdot {K}^{2}, J, -2 \cdot J\right) \cdot 1 \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot {K}^{2}}, J, -2 \cdot J\right) \cdot 1 \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \color{blue}{\left(K \cdot K\right)}, J, -2 \cdot J\right) \cdot 1 \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4} \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

      13. lower-*.f64 50.3

          \[\leadsto \mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, \color{blue}{J \cdot -2}\right) \cdot 1 \]

   8. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \left(K \cdot K\right), J, J \cdot -2\right)} \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto \left(J \cdot \color{blue}{\mathsf{fma}\left(0.25 \cdot K, K, -2\right)}\right) \cdot 1 \]

   if -1e-224 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 66.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 66.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 35.3

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.3%

       \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(-2 \cdot J\_m\right) \cdot t\_0\\ t_2 := t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+217}:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + {\left(\frac{U\_m}{J\_m} \cdot 0.5\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (* (* -2.0 J_m) t_0))
        (t_2 (* t_1 (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_2 (- INFINITY))
      (- U_m)
      (if (<= t_2 5e+217)
        (* t_1 (sqrt (+ 1.0 (pow (* (/ U_m J_m) 0.5) 2.0))))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_2 <= 5e+217) {
		tmp = t_1 * sqrt((1.0 + pow(((U_m / J_m) * 0.5), 2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (-2.0 * J_m) * t_0;
	double t_2 = t_1 * Math.sqrt((1.0 + Math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = -U_m;
	} else if (t_2 <= 5e+217) {
		tmp = t_1 * Math.sqrt((1.0 + Math.pow(((U_m / J_m) * 0.5), 2.0)));
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	t_0 = math.cos((K / 2.0))
	t_1 = (-2.0 * J_m) * t_0
	t_2 = t_1 * math.sqrt((1.0 + math.pow((U_m / ((2.0 * J_m) * t_0)), 2.0)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = -U_m
	elif t_2 <= 5e+217:
		tmp = t_1 * math.sqrt((1.0 + math.pow(((U_m / J_m) * 0.5), 2.0)))
	else:
		tmp = U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(-2.0 * J_m) * t_0)
	t_2 = Float64(t_1 * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_2 <= 5e+217)
		tmp = Float64(t_1 * sqrt(Float64(1.0 + (Float64(Float64(U_m / J_m) * 0.5) ^ 2.0))));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	t_0 = cos((K / 2.0));
	t_1 = (-2.0 * J_m) * t_0;
	t_2 = t_1 * sqrt((1.0 + ((U_m / ((2.0 * J_m) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = -U_m;
	elseif (t_2 <= 5e+217)
		tmp = t_1 * sqrt((1.0 + (((U_m / J_m) * 0.5) ^ 2.0)));
	else
		tmp = U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$2, (-Infinity)], (-U$95$m), If[LessEqual[t$95$2, 5e+217], N[(t$95$1 * N[Sqrt[N[(1.0 + N[Power[N[(N[(U$95$m / J$95$m), $MachinePrecision] * 0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], U$95$m]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 43.3

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 5.00000000000000041e217

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)}}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{U}{J} \cdot \frac{1}{2}\right)}}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{U}{J} \cdot \frac{1}{2}\right)}}^{2}} \]

      3. lower-/.f64 86.5

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\color{blue}{\frac{U}{J}} \cdot 0.5\right)}^{2}} \]

   5. Applied rewrites 86.5%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\color{blue}{\left(\frac{U}{J} \cdot 0.5\right)}}^{2}} \]

   if 5.00000000000000041e217 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 33.7%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 33.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 49.2

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.3%

       \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\_m\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U\_m}{\left(2 \cdot J\_m\right) \cdot t\_0}\right)}^{2}}\\ J\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-U\_m\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{J\_m}, \frac{U\_m \cdot U\_m}{J\_m}, 1\right)} \cdot \left(-2 \cdot J\_m\right)\\ \mathbf{else}:\\ \;\;\;\;U\_m\\ \end{array} \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J_m) t_0)
          (sqrt (+ 1.0 (pow (/ U_m (* (* 2.0 J_m) t_0)) 2.0))))))
   (*
    J_s
    (if (<= t_1 (- INFINITY))
      (- U_m)
      (if (<= t_1 -1e-224)
        (* (sqrt (fma (/ 0.25 J_m) (/ (* U_m U_m) J_m) 1.0)) (* -2.0 J_m))
        U_m)))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J_m) * t_0) * sqrt((1.0 + pow((U_m / ((2.0 * J_m) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U_m;
	} else if (t_1 <= -1e-224) {
		tmp = sqrt(fma((0.25 / J_m), ((U_m * U_m) / J_m), 1.0)) * (-2.0 * J_m);
	} else {
		tmp = U_m;
	}
	return J_s * tmp;
}

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J_m) * t_0) * sqrt(Float64(1.0 + (Float64(U_m / Float64(Float64(2.0 * J_m) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U_m);
	elseif (t_1 <= -1e-224)
		tmp = Float64(sqrt(fma(Float64(0.25 / J_m), Float64(Float64(U_m * U_m) / J_m), 1.0)) * Float64(-2.0 * J_m));
	else
		tmp = U_m;
	end
	return Float64(J_s * tmp)
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U$95$m / N[(N[(2.0 * J$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(J$95$s * If[LessEqual[t$95$1, (-Infinity)], (-U$95$m), If[LessEqual[t$95$1, -1e-224], N[(N[Sqrt[N[(N[(0.25 / J$95$m), $MachinePrecision] * N[(N[(U$95$m * U$95$m), $MachinePrecision] / J$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J$95$m), $MachinePrecision]), $MachinePrecision], U$95$m]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 5.5%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 43.3

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 43.3%

      \[\leadsto \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -1e-224

   1. Initial program 99.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \color{blue}{-2 \cdot \left(J \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot \left(-2 \cdot J\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}} \cdot \left(-2 \cdot J\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{1 + \frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}}}} \cdot \left(-2 \cdot J\right) \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{J}^{2}} + 1}} \cdot \left(-2 \cdot J\right) \]

      6. associate-*r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4} \cdot {U}^{2}}{{J}^{2}}} + 1} \cdot \left(-2 \cdot J\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{4} \cdot {U}^{2}}{\color{blue}{J \cdot J}} + 1} \cdot \left(-2 \cdot J\right) \]

      8. times-frac N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4}}{J} \cdot \frac{{U}^{2}}{J}} + 1} \cdot \left(-2 \cdot J\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{{U}^{2}}{J}, 1\right)}} \cdot \left(-2 \cdot J\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4}}{J}}, \frac{{U}^{2}}{J}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \color{blue}{\frac{{U}^{2}}{J}}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      12. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{\color{blue}{U \cdot U}}{J}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{\frac{1}{4}}{J}, \frac{\color{blue}{U \cdot U}}{J}, 1\right)} \cdot \left(-2 \cdot J\right) \]

      14. lower-*.f64 48.5

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \color{blue}{\left(-2 \cdot J\right)} \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\frac{0.25}{J}, \frac{U \cdot U}{J}, 1\right)} \cdot \left(-2 \cdot J\right)} \]

   if -1e-224 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 66.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 66.6%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 35.3

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 35.3%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.3%

       \[\leadsto U \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U\_m\\ \mathbf{else}:\\ \;\;\;\;-U\_m\\ \end{array} \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m)
 :precision binary64
 (* J_s (if (<= (cos (/ K 2.0)) -0.04) U_m (- U_m))))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = U_m;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.04d0)) then
        tmp = u_m
    else
        tmp = -u_m
    end if
    code = j_s * tmp
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.04) {
		tmp = U_m;
	} else {
		tmp = -U_m;
	}
	return J_s * tmp;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.04:
		tmp = U_m
	else:
		tmp = -U_m
	return J_s * tmp

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = U_m;
	else
		tmp = Float64(-U_m);
	end
	return Float64(J_s * tmp)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp_2 = code(J_s, J_m, K, U_m)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.04)
		tmp = U_m;
	else
		tmp = -U_m;
	end
	tmp_2 = J_s * tmp;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], U$95$m, (-U$95$m)]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 73.3%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. sin-sum N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. flip3-+ N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. Applied rewrites 73.2%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
   6. Step-by-step derivation

      1. cos-neg-rev N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

      16. lower-*.f64 33.0

          \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
   8. Step-by-step derivation

   9. Applied rewrites 33.0%

      \[\leadsto U \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.0%

       \[\leadsto U \]

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 72.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-1 \cdot U} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(U\right)} \]

      2. lower-neg.f64 23.1

         \[\leadsto \color{blue}{-U} \]

   5. Applied rewrites 23.1%

      \[\leadsto \color{blue}{-U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 14.0% accurate, 373.0× speedup

Math

\[\begin{array}{l} U_m = \left|U\right| \\ J\_m = \left|J\right| \\ J\_s = \mathsf{copysign}\left(1, J\right) \\ J\_s \cdot U\_m \end{array} \]

FPCore

U_m = (fabs.f64 U)
J\_m = (fabs.f64 J)
J\_s = (copysign.f64 #s(literal 1 binary64) J)
(FPCore (J_s J_m K U_m) :precision binary64 (* J_s U_m))

C

U_m = fabs(U);
J\_m = fabs(J);
J\_s = copysign(1.0, J);
double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Fortran

U_m =     private
J\_m =     private
J\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j_s, j_m, k, u_m)
use fmin_fmax_functions
    real(8), intent (in) :: j_s
    real(8), intent (in) :: j_m
    real(8), intent (in) :: k
    real(8), intent (in) :: u_m
    code = j_s * u_m
end function

Java

U_m = Math.abs(U);
J\_m = Math.abs(J);
J\_s = Math.copySign(1.0, J);
public static double code(double J_s, double J_m, double K, double U_m) {
	return J_s * U_m;
}

Python

U_m = math.fabs(U)
J\_m = math.fabs(J)
J\_s = math.copysign(1.0, J)
def code(J_s, J_m, K, U_m):
	return J_s * U_m

Julia

U_m = abs(U)
J\_m = abs(J)
J\_s = copysign(1.0, J)
function code(J_s, J_m, K, U_m)
	return Float64(J_s * U_m)
end

MATLAB

U_m = abs(U);
J\_m = abs(J);
J\_s = sign(J) * abs(1.0);
function tmp = code(J_s, J_m, K, U_m)
	tmp = J_s * U_m;
end

Wolfram

U_m = N[Abs[U], $MachinePrecision]
J\_m = N[Abs[J], $MachinePrecision]
J\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[J$95$s_, J$95$m_, K_, U$95$m_] := N[(J$95$s * U$95$m), $MachinePrecision]

Derivation

1. Initial program 73.0%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. sin-+PI/2-rev N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\sin \left(\frac{K}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. sin-sum N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   4. flip3-+ N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3} + {\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)}^{3}}{\left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \left(\sin \left(\frac{K}{2}\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

4. Applied rewrites 72.8%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \color{blue}{\frac{{\left(0 \cdot \sin \left(\frac{K}{2}\right)\right)}^{3} + {\left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)}^{3}}{\mathsf{fma}\left(0 \cdot \sin \left(\frac{K}{2}\right), 0 \cdot \sin \left(\frac{K}{2}\right), \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right) - \left(0 \cdot \sin \left(\frac{K}{2}\right)\right) \cdot \left(1 \cdot \cos \left(\frac{K}{-2}\right)\right)\right)}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

5. Taylor expanded in U around -inf

   \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{1}{2} \cdot K\right)}} \]
6. Step-by-step derivation

   1. cos-neg-rev N/A

      \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}} \]

   2. distribute-lft-neg-in N/A

      \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot K\right)} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{U \cdot \cos \left(\frac{-1}{2} \cdot K\right)}{\cos \left(\frac{-1}{2} \cdot K\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(\frac{-1}{2} \cdot K\right) \cdot U}}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   7. cos-neg-rev N/A

      \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   8. lower-cos.f64 N/A

      \[\leadsto \frac{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   9. distribute-lft-neg-in N/A

      \[\leadsto \frac{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot U}{\cos \left(\frac{-1}{2} \cdot K\right)} \]

   12. cos-neg-rev N/A

       \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

   13. lower-cos.f64 N/A

       \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)}} \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)}} \]

   15. metadata-eval N/A

       \[\leadsto \frac{\cos \left(\frac{1}{2} \cdot K\right) \cdot U}{\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right)} \]

   16. lower-*.f64 31.1

       \[\leadsto \frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \color{blue}{\left(0.5 \cdot K\right)}} \]

7. Applied rewrites 31.1%

   \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot K\right) \cdot U}{\cos \left(0.5 \cdot K\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 31.1%

   \[\leadsto U \cdot \color{blue}{1} \]
10. Step-by-step derivation

11. Applied rewrites 31.1%

    \[\leadsto U \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))